The mid-segment of a triangle is a segment joining the of two sides of a triangle.

Transcription

1 5.1 and 5.4 Perpendicular and Angle Bisectors & Midsegment Theorem THEOREMS: 1) If a point lies on the perpendicular bisector of a segment, then the point is equidistant from the endpoints of the segment. Write the converse: 2) If a point is equidistant from the sides of an angle of a triangle, then the point lies on the bisector of the angle. Write the converse: The mid-segment of a triangle is a segment joining the of two sides of a triangle. Properties of a mid segment: 1. is to the third side 2. is as long as the third side. Example 1: Use the points A(2,2) B(12,2), and C(4,8) for the following. 1) Find X and Y, the midpoints of AC and CB. 2) Find XY and AB ) Find the slope of AB and XY 4) What is the slope of a line parallel to x + 2y = 12? Z M, N, and P are midpoints of XZ,ZY, and XY, respectively. 1.) Mark the diagram with tick marks: 2) Name all s: ) XY // ; XZ // ; MP // X M P N Y Example 1) Given DE, DF, and FE are the lengths of Example 2) Given AC = 42, CB = 46, mid-segments. Find the perimeter of triangle ABC. AB = 48, D, E, and F are midpoints Find the perimeter of triangle DEF Example ) D and E are midpoints. Find m<a and Example 4) Find the value of x. The diagram m< EDA. Is NOT drawn to scale. 1

2 Example 5) Find the value of x. Example 6) Points B, D, and F are midpoints. EC = 0 and DF = 2. Find AC. Example 7) Identify the mid-segment and find its length. Example 8) If BE = 2x+6 and DF = 5x+9, find the value of x, DF, and BE Example 9) Q is equidistant from the sides of the value of x. The diagram is not to scale. Find Use the figure for Exercises Given that line p is the perpendicular bisector of XZ and XY 15.5, find ZY.. Given that XZ 8, YX 27, and YZ 27, find ZW. 4. Given that line p is the perpendicular bisector of XZ ; XY 4n, and YZ 14, find n. 5. Given that XY ZY, WX 6x 1, and XZ 10x 16, find ZW. Use the figure for Exercises Given that FG HG and m FEH 558, find m GEH. 7. Given that EG bisects FEH and GF 2, find GH. 8. Given that FEG GEH, FG 10z 0, and HG 7z 6, find FG. 9. Given that GF GH, m GEF 8 a8, and m GEH 248, find a. 2

3 Many plants grow in geometric patterns. The figure shows the veins in a leaf from an alder tree. Refer to the figure for Exercises 2 5. Match the letter of each theorem to the statement that uses the theorem. 2. If BD CD, then D is on the bisector of BAC.. If BAD CAD, then BD CD. 4. If QP RP and SP QR, then QS RS. 5. If QS RS and QP RP, then SP QR. A. Perpendicular Bisector Theorem B. Converse of the Perpendicular Bisector Theorem C. Angle Bisector Theorem D. Converse of the Angle Bisector Theorem Use the figure for Exercises 6 and Given that line m is the perpendicular bisector of FH and EH 100, find EF. 7. Given that EF 1, FH 10, and EH 1, find GH. Use the figure for Exercises 8 and Given that JL bisects KJM and KL 42, find ML. 9. Given that KL 4 and ML 4 and m MJL 408, find m KJL.

4 5.2 and 5. Special Segments of a : Def: A median of a triangle is Draw the three medians of the following triangle: The point of concurrency of the three medians of a triangle is called the. Def : An altitude of a triangle is Draw the altitude from all three vertices: The point of concurrency of the lines containing the altitudes of a triangle is called the. Def An angle bisector is Draw the angle bisectors of all vertices: The point of concurrency of the < bisectors of the angles of a triangle is called the. Def Perpendicular bisector of a segment:. Draw the perpendicular bisectors of each side of the : The point of concurrency of the bisectors of the sides of a triangle is called the ***Note: Angle bisectors, medians and altitudes always have a vertex as one endpoint. This is not always the case for perpendicular bisectors. 4

5 Centroid Theorem The medians of a are concurrent at a point that is 2/ the distance from each vertex to the midpoint of the opposite side. Example 1) D is the centroid of ABC and DE = 6. Find BD and BE. If AD = 9, find DF and AF. Example 2) name the median, altitude, and angle bisector 5

6 6

7 Practice with Altitudes and Medians Each figure below shows one or more medians. 1. Find x. DB is a median. SU is a median. Find x. In PRS, PT is an altitude and PX is a median. 2. Find RS if RX = x + 7 and SX = x 11.. Find RT if RT = x 6 and m PTR = 8x Find x if EG is a median of DEF. Find the coordinates of the centroid of each triangle. 5. 7

8 Find the value of each variable. 1) 2) ) 4) 5) Find the coordinates of the centroid of each triangle given the three vertices. 6) 7) 8

9 5.5 : and 5.6 Inequalities (1 and 2 Triangles) In a, the smallest is opposite the shortest side. 54 D 12mm In a, the largest is opposite the longest side. 1mm E Converses are also true! 52 11mm In a, the shortest side is opposite the smallest. B C F In a, the longest side is opposite the largest. 1. In ABC name the sides in order from least to greatest. 2. In DEF name the angles in order from greatest to least.. Name the shortest and longest sides in right FIT if F is the right angle and m I = 48. Theorem The sum of the lengths of any 2 sides of a triangle is greater than the length of the rd side. 8. Can a triangle have sides with the given lengths? a) 4m, 7m, 8m d) 1.2cm, 2.6cm, 4.9cm b) 4in, 4in, 4in e) 11m, 12m, 14m c) 18ft, 20ft, 40ft f) 2.5m,.5m, 6m TRIANGLE INEQUALITIES II. In the following exercises the diagrams are not drawn to scale. If each diagram were drawn to scale, which numbered angle would be largest? Which segment would be x+ 1 the largest? 1 12 x x - 1 The lengths of two sides of a triangle are given. Find the range of possible lengths for the third side m,.5 m ft, 177 ft mi, 4 mi 9

10 The diagrams are not drawn to scale. Which numbered angle would be the largest? cm 6 cm cm y 1 y y + 2 Which segment is the longest? B C 6 60 A 58 Use lengths to complete: 9. B (x + 1) (x + ) A C a 61 b > > 59 c 10

11 Compare the given measures. 1. m K and m M 2. AB and DE. QR and ST Find the range of values for x